M2. REAL ANALYSIS
Course coordinator: Laszlo Csirmaz
No. of Credits: 3, and no. of ECTS credits: 6
Time Period of the course: Fall Semester
Prerequisites: Undergraduate calculus, Elementary Linear Algebra
Course Level: introductory MS
Brief introduction to the course:
Introduction to Lebesgue integration theory; measure, σ-algebra, σ-finite measures. Different
notion of convergences; product spaces, signed measure, Radon-Nikodym derivative, Fubini and
Riesz theorems; Weierstrass approximation theorem. Solid foundation in the Lebesgue
integration theory, basic techniques in analysis. It also enhances student’s ability to make their
own notes.At the end of the course students are expected to understand the difference between
”naive” and rigorous modern analysis. Should have a glimpse into the topics of functional
analysis as well. They must know and recall the main results, proofs, definition.
The goals of the course:
The main goal of the course is to introduce students to the main topics and methods of Real
Analysis.
The learning outcomes of the course:
By the end of the course, students areexperts on the topic of the course, and how to use these
methods to solve specific problems. In addition, they develop some special expertise in the topics
covered, which they can use efficiently in other mathematical fields, and in applications, as well.
They also learn how the topic of the course is interconnected to various other fields in mathematics,
and in science, in general. At the end of the course students are expected to understand the
difference between ”naive” and rigorous modern analysis. They should have a glimpse into the
topics of functional analysis as well. They must know and recall the main results, proofs,
definition.
More detailed display of contents (week-by-week):
1. Outer measure, measure, σ-algebra, σ-finite measure. liminf and limsup of sets; their measure.
The Borel-Cantelli lemma. Complete measure
2. Caratheodory outer measure on a metric space. Borel sets. Lebesgue measure. Connection
between Lebesgue measurable sets and Borel sets
3. Measurable functions. Measurable functions are closed under addition and multiplication.
Continuous functions are measurable. Example where the composition of measurable functions
is not measurable
4. Limits of measurable functions, sup, inf, lim sup, lim inf. Egoroff's theorem: if fi converges
pointwisea.e to f then it converges uniformly with an exceptional set of measure <ε.
Convergence in measure; pointwise convergence for a subsequence.
5. Lusin's theorem: a Lebesgue measurable function is continuous with an exceptional set of
measure <ε. Converging to a measurable function by simple functions.
6. Definition of the integral; conditions on a measurable function to be integrable. Fatou's
lemma, Monotone Convergence Theorem; Lebesgue's Dominated Convergence Theorem.
Counterexample: a sequence of functions tends to f, but the integrals do not converge to the
integral of f.
7. Hölder and Minkowsi inequalities; Lp is a normed space.Riesz-Fischer theorem: Lp is
complete, conjugate spaces, basic properties
8. Signed measure, absolute continuity, Jordan and Hahn decomposition. RadonNikodymderivative.Product measure, Fubini's theorem. Counterexample where the order of
integration cannot be exchanged
9. Example for a continuous, nowhere differentiable function. Example for a strictly increasing
function which has zero derivative a.e.
10. An increasing function has derivative a.e.
11. Weierstrass' approximation theorem
12. Basic properties of convolution
References:
Online material is available at the following sites:
1. http://www.indiana.edu/~mathwz/PRbook.pdf,
2. http://compwiki.ceu.hu/mediawiki/index.php/Real_analysis